Theresa May’s Brexit bill has cleared all its hurdles in the Houses of Parliament, opening the way for the prime minister to trigger article 50 by the end of March.

Peers accepted the supremacy of the House of Commons late on Monday night after MPs overturned amendments aimed at guaranteeing the rights of EU citizens in the UK and giving parliament a “meaningful vote” on the final Brexit deal. The decision came after a short period of so-called “ping pong” when the legislation bounced between the two houses of parliament as a result of disagreement over the issues.

The outcome means the government has achieved its ambition of passing a “straightforward” two-line bill that is confined simply to the question of whether ministers can trigger article 50 and start the formal Brexit process.

It had been widely predicted in recent days that May would fire the starting gun on Tuesday, immediately after the vote, but sources quashed speculation of quick action and instead suggested she will wait until the final week of March.

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MPs voted down the amendment on EU nationals’ rights by 335 to 287, a majority of 48, with peers later accepting the decision by 274 to 135. The second amendment on whether to hold a meaningful final vote on any deal after the conclusion of Brexit talks was voted down by 331 to 286, a majority of 45, in the Commons.

The Lords then accepted that decision by 274 to 118, with Labour leader Lady Smith telling the Guardian that continuing to oppose the government would be playing politics because MPs would not be persuaded to change their minds. “If I thought there was a foot in the door or a glimmer of hope that we could change this bill, I would fight it tooth and nail, but it doesn’t seem to be the case,” she said.

But the decision led to tensions between Labour and the Lib Dems, whose leader, Tim Farron, hit out at the main opposition. “Labour had the chance to block Theresa May’s hard Brexit, but chose to sit on their hands. Tonight there will be families fearful that they are going to be torn apart and feeling they are no longer welcome in Britain. Shame on the government for using people as chips in a casino, and shame on Labour for letting them,” he said.

The amendments would have required the government to bring forward proposals about how they would protect EU citizens within three months of triggering article 50, and said that a parliamentary vote on the final Brexit deal should be on the face of the legislation.

The Brexit secretary, David Davis, told colleagues that MPs and peers had made their arguments with “passion, sincerity and conviction”. But, using emollient language that served to persuade peers not to cause any more trouble for the government, he said he wanted this legislation to remain “straightforward”, simply allowing the government to embark on the formal Brexit process.

Davis said he would take personal “moral responsibility” for guaranteeing the future rights of EU citizens in the UK as well as those of Britons living on the continent.

David Davis opens the debate on the Brexit bill on Monday. Photograph: AFP/Getty Images

On the second amendment, Davis said guaranteeing a meaningful vote could hamper the government during its negotiations. He questioned the motives of those arguing for it, claiming that they wanted to reverse the referendum result.

A number of Tory MPs argued that the government was right to aim to guarantee reciprocity for British citizens abroad. However, Davis was opposed by Labour, the Lib Dems and the SNP – with some passionate speeches from critics – and faced a small rebellion on his own backbenches over the meaningful vote on the eventual Brexit deal.

Davis reassured MPs that they would have a say on various issues through government bills that would be published after the great repeal bill including on immigration and Britain’s customs deal. It came as reports suggested that a leaked document had outlined how those bills would be among seven pieces of legislation to prepare for life outside the EU.

After peers voted to allow the passage of the bill unamended, Davis said the decision had placed Britain “on the threshold of the most important negotiation for our country in a generation”, saying that his government was ready to trigger article 50 and forge new trade links.

The Labour leader, Jeremy Corbyn, described the decision by MPs to overturn the votes as “deeply disappointing”. He said: “We will continue to demand that the stress they, and British citizens living in the EU, are being put under is ended, and they are given the right to remain.

“Article 50 is being triggered because of the result of the EU referendum. But it is only the start of the process. Labour, at every stage, will challenge the government’s plans for a bargain-basement Brexit with Labour’s alternative of a Brexit that puts jobs, living standards and rights first.”

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The founder of The3million, the grassroots organisation lobbying for the rights of EU citizens, said he felt “utter desperation” that they are now destined to become bargaining chips.

Nicolas Hatton said: “The hearts of 3 million EU citizens living in the UK will have sunk today when they heard that MPs had voted down the amendment to article 50 giving them guarantees. This was the last chance, and I struggle to find words to express my utter desperation that EU citizens will now be used by the government as bargaining chips in the Brexit negotiations.”

There was also angry reaction from British people living on the continent. Dave Spokes for Expat Citizen Rights in EU, one of the biggest support groups, with more than 7,600 members in 27 EU countries, said: “It is worrying that our government chooses to ignore the concerns of its own citizens and the evidence put to its select committees that citizens’ rights should be confirmed immediately.

“The government’s own white paper said it had engaged with citizens’ groups in Europe, but we have yet to find one group that has been approached by the Department for Exiting the EU. We do wonder what the outcome might have been had they actually done so.”

A coalition of 11 grassroots groups campaigning for British nationals in the EU said it also felt for the millions of Europeans in the UK. “We share their suffering, and know exactly how stressful and unpleasant it is to live with this degree of uncertainty for ourselves and our families,” said spokeswoman Jane Golding, who lives in Germany. “We do not believe people should be used as a bargaining chips.”